Question: We have a triangle $\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$?
Let's draw a sketch first. Since $\triangle ABC$ is isosceles, we know that $AM$ must form a right angle with $BC.$ [asy]
pair A, B, C, M;
A = (0, 6.24);
B = (-5, 0);
C = (5, 0);
M = 0.5 * B + 0.5 * C;
draw(A--B--C--cycle);
draw(A--M);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$M$", M, S);
draw(rightanglemark(A,M,B,10));
[/asy] We know that $BM = MC = \frac{BC}{2} = 5.$ Now we just simply apply the Pythagorean Theorem on the right triangle $\triangle ABM.$ \begin{align*}
AM^2 &= AB^2 - BM^2\\
AM^2 &= 8^2 - 5^2 = 39\\
AM &= \boxed{\sqrt{39}}
\end{align*}